Hyperbolic function

Hyperbolic functions are combinations of the exponential function analogous to trigonometric functions. The hyperbolic cosine and hyperbolic sine satisfy ${\cosh }^{2}x-{\sinh }^{2}x=1$ and parametrize a unit hyperbola.

The hyperbolic cosine is $\cosh x=(e^x+e^{-x})/2$.

For $x\ge 1$, the inverse hyperbolic cosine on its nonnegative real branch is $\mathrm{arcosh}x=\log (x+\sqrt{x^2-1})$.

The hyperbolic sine is $\sinh x=(e^x-e^{-x})/2$.

The hyperbolic cotangent is $\coth x=\cosh x/\sinh x$ where $\sinh x\ne 0$.

The hyperbolic secant is